Absolute value

Definition

Let $x$ be a real number. The absolute value of $x$ is the real number $$|x| = \begin{cases} x & \text{if } x \ge 0, \\ -x & \text{if } x < 0. \end{cases}$$

Remarks

For each real $x$: $|x| \ge 0$; $|x| = 0$ only for $x = 0$; $|-x| = |x|$; and $|x| = \max(x, -x)$. The distance between two real numbers $x$ and $y$ is $|x - y|$. For $r \ge 0$, the inequality $|x| \le r$ holds exactly when $-r \le x \le r$.

Explanations

L1 human

The absolute value removes the minus sign from a number. On the number line, $|x|$ is the distance from $x$ to $0$, and $|x - y|$ is the distance from $x$ to $y$.